Part $1$: Duration and Convexity Calculations

Consider a bond with the following terms:

$10$ years to maturity

$$ $$1,000$ face value

$$ Coupons are paid $2$ times per year

$$ Annual coupon rate is $5\%$

For problems $1-5,$ assume a constant discount rate across maturities of $6\%.$ Also, assume that the bond will make its next coupon payment in exactly $1/2$ years.

$1.$ Find the current price of the bond

$2.$ Compute the exact modified duration of the bond $($not the approximation$)$

$3.$ Approximate the modified duration of the bond

$4.$ Compute the exact convexity of the bond $($not the approximation$)$

$5.$ Approximate the convexity of the bond